SCIENTIFIC PROGRAMS AND ACTIVITIES

March 20, 2018

FIELDS
UNDERGRADUATE SUMMER RESEARCH PROGRAM

July
to August, 2016

Application
to participate now open! See Application Process for details.

The
Fields Institute will host the Fields Undergraduate Summer Research
Program in July and August of 2016 in Toronto. The Program supports
up to 25 students in mathematics-related disciplines to participate
in research projects supervised by leading scientists from Fields
Institute Thematic & Focus Programs or partner
universities.

Students
who are accepted into the Program will be receive a daily meal
allowance and, depending on where they are travelling from, the
following additional support:

Students
from within the Greater Toronto Area (GTA) will be reimbursed
for the cost of a monthly public transit pass (TTC
Metropass).

Students
from outside the GTA will receive financial support for
travel to and residence in Toronto (student residence housing
at the University of Toronto Downtown (St. George) Campus)
for the duration of the Program, and

Students
from outside of Canada will be provided the same support
as students from outside the GTA, plus medical coverage during
their stay.

Students
will work on research projects in groups of 3 to 5.

Supervisors
may suggest other topics for students in addition to the research
projects outlined below. Students may also have the opportunity
to visit the home campus of the their supervisors to get to know
their universities.

During
the application process, you will be prompted to upload your Curriculum
Vitae (CV) together with a letter outlining your relevant
background and experience. Your letter must not exceed two letter-sized
pages, and must be in 12-point Times New Roman font, with single
line spacing. Top, bottom, and side margins each must be no less
than one inch. Your letter and CV must be submitted in a single,
combined PDF document.

As
part of the online application, unofficial transcipts may be submitted
for initial review (formatted as PDF document). This step is not
mandatory, but it is recommended. Please note that any unofficial
transcripts submitted will be matched against official transcripts
once received.

For
your application to be considered, you must also arrange to have
the following documents provided directly to the Fields Institute,
on or before the application deadline noted below:

1.
Official transcripts* from your home university should be addressed
to:

Fields
Undergraduate Summer Research Program
Fields Institute for Research in Mathematical Sciences
222 College Street
Toronto, ON
M5T 3J1 Canada

*An
official transcript is prepared and sent by the issuing school
usually by the Student Registrar with an original signature
of a school official. Source

2.
Two letters or statements of reference from someone who can provide
a candid evaluation of your qualifications or skills. You must
provide a minimum of 2 references, however, there is space on
the online application to provide 3. The referee will be contacted
using the details supplied on the application form; therefore,
be sure to enter their information correctly. They will be asked
to submit their letters confidentially online as text or a pdf
file. It is the responsibility of the student to ensure that reference
letters have been submitted by the deadline below.

To
be considered for the Program, all of your materials must be received
by the Fields Institute before11:59 pm Eastern Standard
Time on February 29, 2016.

Accepted
students requiring visas for travel to Canada will need to make
their own arrangements to obtain the necessary documents.

The
motivation comes from a toy model for quantum gravity. It is
known that the fabric of spacetime in very short distances is
not classical anymore and should be replaced by a hitherto mysterious
quantum spacetime. At the same time one needs to take into account,
that is to integrate over, all possible geometries. Quantum
gravity is ''terra incognita", but in the present model
different geometries, that is different metrics, can be parametrized
by spaces of self-adjoint matrices. All integrals are finite
dimensional and well defined. One goal is to understand if there
are any kind of universal laws governing the distribution of
eigenvalues as the size of matrices grow. While there are some
similarities with random matrix theory, the nature of the current
project is quite different and the subject is still in its infancy.
Most of the integrals will be computed by computer simulation
and Monte Carlo methods, but a theoretical understanding of
them would be very important. A good undergraduate math education
is enough to handle this project.

A
Riemann surface is a two dimensional space (like a sphere or
the surface of a donut) with a notion of angle between tangent
vectors. The amount by which angles have to be distorted to
get from one Riemann surface to another defines a distance on
the space of all Riemann surfaces of a given topological type,
called the Teichmuller metric. The most efficient way to get
from one Riemann surface to another in this metric can be described
in terms of flat structures. A flat structure is a way to assemble
the surface from polygons in the plane by gluing pairs of parallel
edges via isometries. One can deform a flat structure by stretching
it in a fixed direction. The geodesics (or straight lines) in
the Teichmuller metric are given precisely by these stretch
paths. Despite this concrete description, the geometry of this
metric is still poorly understood. For example, it was discovered
only recently that balls in this metric are not necessarily
convex, at least when the topology of the surface is sufficiently
complicated. The goal of the project is to study the low complexity
cases that remain, starting with the sphere with 5 points removed.
The problem is amenable to numerical computations.

Description:
Combinatorial models for concurrent systems were used by Jardine
and his group at Western to construct algorithms for identifying
execution paths. These algorithms study an entire model at once
and thus work well only for small examples, since exponential
complexity levels can be involved. The algorithms have also
resisted parallelization.

These
models admit patching, meaning that one can construct a model
from smaller pieces and the intersections of those pieces. The
invariant (the "path category") from which one recovers
execution paths respects such patches, in a rather opaque theoretical
way. Generally, one wants to find an algorithm for recovering
the path category of a concurrent system from path categories
for a covering family of patches.

The
purpose of this project is more modest: to find an algorithm
for constructing the path category for a union of two subsystems,
where the path categories of the two subsystems and their intersection
are known. We hope to produce an algorithm and code that solves
this problem.

A
solution of the two-subobject patching problem would point to
parallelization techniques for the analysis of large structures.

The
two-subobject patching problem is simple enough that it can
be explained to upper-year undergraduates. What is really needed
at this stage for this project is a group of researchers with
a bit of algebraic sophistication and programming skills, that
can work collaboratively on this problem.

Project
4: The Mathematics of GlassSupervisor:
C. Sean Bohun (UOIT)

Glass is ubiquitous in contemporary society. In fact, we take
this fascinating material for granted without giving a thought
to how it is manufactured. In some situations it can be treated
as a solid while in others, a viscous fluid. Problems concerning
the manufacture of drawn glass fibres or the large thin sheets
require a blending of analysis, asymptotic methods and numerics,
all working in concert, to discern the underlying behaviour.
For the last 20 years there has been a concerted effort to develop
appropriate models for the glass industry to help them deal
with increasingly restrictive tolerances.

Throughout the project a number of modern models will be considered
and the background mathematics developed as required. We will
consider in particular models of drawing glass into fibres,
the formation of sheets and some of the current pressures faces
by the glass industry. Methods of modelling classical solids,
and linear elastodynamics will form the foundations of the material
which will then be used to develop approximate theories and
analyzed with a asymptotic and numerical methods that are tuned
to the model at hand.

In addition to this, some emphasis will be made concerning
the overarching view of information transfer. In particular
development of the skills to: (i) 'find the mathematics' within
a problem from the glass industry; and (ii) translate mathematical
insights of the problem into focussed expertise that is explained
in a non-technical way.

Program

Activities start the week of July 4, 2016 at the Fields Institute,
222 College Street, Toronto, ON. Map
to Fields

If you are coming from the Woodsworth
College Residence, walk south on St. George Street to College
Street, turn right, Fields is the second building on your right.

Week 1

Day 1

Introductory
Session: Introduction and presentation of the program (introduction
to supervisors, and overview of theme areas and projects).
Student/Supervisor introductions and networking. Lunch provided.

Orientation
Meeting: Students meet with Fields program staff to discuss
computer accounts, offices, expense reimbursements, and
overview of Fields facilities.

Days 2-5

Students
meet informally with supervisors and in their groups to
work on research project.

Week 2

Students
meet informally with supervisors and in their groups to work
on research project.

Week 3

Students
meet informally with supervisors and in their groups to work
on research project.

Introduction
to the Fields SMART board and video conferencing facilities
which are useful for remote collaboration.

Weeks 4 - 8

Students
meet informally with supervisors and in their groups to work
on research project.

Group
excursion (all students welcome). Organized and sponsored by
the Fileds Institute.

Week 9

Mini-Conference:
Students present project results to other supervisor/student
teams.

Students
prepare project report and narrative about their experience
in the Program.
Reports are due on August 31.

1) Print out the form and fill it in as described in the steps
below
2) Only fill out section 1 and then sign and date section 4 on
the second page
3) The Members effective date of coverage is
the date you arrived to Toronto
4) Please leave the Member ID # and Number of
Months of Coverage blank
5) Scan the form and email it as a PDF to cpe@fields.utoronto.ca

Frequently
Asked Questions

I am planning to graduate this upcoming June and I was wondering
if I am still eligible to participate in this program?

Yes,
but preference is given to students going into their final
year or earlier.

Is
there a GPA requirement for students to apply?

No,
but students with higher GPA rank higher during the selection
process.

Are students without prior research experience in a Mathematical
discipline, but with experience in, for example, eligible for
the Program?

Yes,
we welcome students with experience in any area of mathematical
sciences.

Can
the references be of character in nature?

Letters
should address the academic and research backgrounds of
the applicant as much as possible, in addition to character
references if deemed relevant for the program.

I
am interested in this year's research problems, but I am a graduate
or postdoctoral level student. Can I still apply?

This
Program is intended for undergraduate level students: if you
have already completed an undergraduate degree in mathematics,
you are not eligible for the Program.